Sunlight provides an abundant, renewable, accessible source of naturally occurring rays that, for all intents and purposes, are parallel. It is the rare example of a real-world context with which nearly all children are familiar. Despite its familiarity and near universality, sunlight plays almost no part in K-12 mathematics classrooms. This is a missed opportunity in geometry classrooms in particular, where sunlight could provide not only an exemplification of the Euclidean notion of parallel lines but also raw material for doing mathematical work. This note explores the parallelism of sunlight and proposes a collaborative activity that would recreate an ancient geometric triumph: Eratosthenes’ measurement of the circumference of the Earth. 

On the parallelism of sunlight

Eratosthenes of Cyrene was the chief librarian at the library of Alexandria during the 3^rd century BC. Popular retellings of Eratosthenes’ method for measuring the circumference of the earth tend to dwell on the assumption that sun rays are parallel. For example, a recent (2019) article in Medium describes this assumption as, “incorrect, but acceptable for the equipment available at the time.” It is true that Eratosthenes made an assumption about sunlight, but to dismiss this as something that was harmless but ultimately incorrect elides an opportunity for mathematical investigation. 

The parallelism of sunlight can be understood in comparison to a more familiar real world example of parallel lines: railroad tracks. For high-speed rail, the gauge variation, or the allowable variation in the distance between tracks, as measured along any 3m length of railway, must not exceed 6mm. Thus, railroad tracks are permitted to deviate from parallel by nearly 7’ of arc (where 1’ of arc is 1/60 of a degree): 

tan^-1 (6/3000) = .1145 degrees = 6.87’ of arc

This is the angle that is subtended by one 3m (3000mm) length of track and the 6mm tolerance. By contrast, the deviation from parallel of sunlight over a distance of 3m is effectively zero, because Earth is nearly 150 billion meters from the sun: 

tan^-1 (3/150,000,000,000) = .000000001 degrees = .00000006’ of arc

Thus, locally, sun rays are not only more parallel than railroad tracks, their deviation from parallel is well below the threshold of what can be measured with a theodolite. Even globally, sunlight’s deviation from parallel is all but negligible: 

tan^-1 (6,500,000/150,000,000,000) = .0025 degrees = .15’ of arc

So, it is technically true, in a narrow sense, that sun rays striking the Earth are not parallel. But practically, effectively, sun rays are parallel, and this parallelism is a resource that can be harnessed for doing mathematical work. 

The solstice, the cities, and the well

Apart from a concern about the parallelism of sun rays, other legacies of the legend of Eratosthenes are the significance of the date—the measurements are said to have occurred on the summer solstice, the locations of the cities in which the measurements were taken—Syene and Alexandria, and the well that was said to perfectly reflect the noon sun. Narratively, these details add richness to the story that help to situate the measurement as a historically significant achievement that was a product of the ancient world. And it was all those things. However, mathematically, these details bury the lede, which is that the parallel rays of the sun allowed a surface dweller to measure the size of the very earth under his feet; because the sun’s rays are parallel, any observed differences in the altitude of the sun must be a result of the curvature of the earth. Thus, by comparing the altitude of the sun at different locations, it is possible to deduce the size of the earth. The date, the locations, and the well are immaterial. The measurements could take place on any day, anywhere, provided the different places where the measurements take place are not too close together. To recreate the feat, all that is needed is coordination—of the date on which to make the measurements—and improvisation—of a device for measuring the altitude of the sun. 

Here is how it would work: 

1. We choose a date to take the measurements. This could be before the end of the academic year or else in the late summer or early fall, as an activity that could kick off the next academic year. 
2. We engineer devices that can measure the maximum altitude of the sun. The image below (Figure 1) shows an example of one such device, improvised from a phone, a shoe, a pencil, a rubber band, and graph paper. 
3. We compare our measurements against the north/south displacement of where the measurements were taken. For example, if measurement locations are near odd-numbered interstate highways, it may be possible to approximate the north/south displacement from mile markers.  

The device shown in Figure 1 is designed to record the shadows cast by a gnomon as the sun moves across the sky near local apparent noon. The pencil acts as a gnomon, which is the technical name for the part of a sundial that casts a shadow. The shadow of the gnomon is projected onto graph paper, which allows for the calculation of the apparent altitude of the sun (using the height of the gnomon off the ground, the length of its shadow, and the arctangent relationship). Finally, the apparatus is mounted to a phone, so that the movement of the shadow around solar noon could be recorded. When the shadow reaches its shortest length, the sun attains its highest altitude, and that is the measurement that would be used for the activity. This is intended only as an example of a device that could be improvised; readers are encouraged to use their own design and engineering skills to create something better.

This is a sketch of a plan that will be hashed out in more detail among the group of readers (and their students!) that want to participate in this collaborative recreation of Eratosthenes’s feat. If you would like to participate, please email me (on or before Monday, June 7) at ju***********@***ne.edu. I will email those who express interest, and we will come up with a plan that works for everyone who wants to be involved.

References

Decamp, N., & Hosson, C. (2012). Implementing Eratosthenes’ discovery in the classroom: Educational difficulties needing attention. Science & Education, 21(6), 911-920.